THEORY OF LANGUAGE SYNTAX

Nijhoff International Philosophy Series

VOLUME 42

General Editor: JAN T . J . SRZEDNICKIEditor for volumes on Logic and Applying Logic : STANISLAW J . SURMAEditor for volumes on Contributions to Philosophy : JAN T . J . SRZEDNICKIAssistant to the General Editor : DAVID WOOD

Editorial Advisory Board:

R.M. Chisholm (Brown University, Rhode Island) ; Mats Furberg (Goteborg Univer-sity) ; D.A.T. Gasking (University of Melbourne) ; H.L.A. Hart (University College,Oxford) ; S. Korner (University of Bristol and Yale University) ; H .J . McCloskey (LaTrobe University, Bundoora, Melbourne); J . Passmore (Australian National Univer-sity, Canberra) ; A. Quinton (Trinity College, Oxford) ; Nathan Rotenstreich (TheHebrew University, Jerusalem) ; Franco Spisani (Centro Superiore di Logica e ScienzeComparate, Bologna); R. Ziedins (Waikato University, New Zealand)

The titles published in this series are listed at the end of this volume.

Urszula Wybraniec-Skardowska

Theory ofLanguage Syntax

Categorial Approach

W K A P - A R C H I E F

I NMKLUWER ACADEMIC PUBLISHERSDORDRECHT / BOSTON / LONDON

Library of Congress Cataloging-in-Publication Data

Wybraniec-Skardowska, Urszula .[Teorie ,igzykow syntaktycznie kategorialnych . English]Theory of language syntax : categorial approach / by Urszula

Wybraniec-Skardowska .p . cm . -- (Nijhoff international philosophy series : v . 42)

Translation of : Teorie ,igzykow syntaktycznie kategorialnych .Includes bibliographical references and index .ISBN 0-7923-1142-61 . Categorial grammar . 2 . Formal languages . 3 . Logic, Symbolic

and Mathematical . I . Title . II . Series .P161 .W913 1991415--dc2O 91-64

ISBN 0-7923-1142-6

Published by Kluwer Academic Publishers,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Kluwer Academic Publishers incorporatesthe publishing programmes ofD. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press .

Sold and distributed in the U.S .A. and Canadaby Kluwer Academic Publishers,101 Philip Drive, Norwell, MA 02061, U.S.A .

In all other countries, sold and distributedby Kluwer Academic Publishers Group,P.O. Box 322, 3300 AH Dordrecht, The Netherlands .

Printed on acidfree paper

This book has originally been published in Polish, entitledTheorie Jgzyk6w Syntaktycznie Kategorialnyckwith PWN, Warsaw, 1985.

Translated from the Polish by Olgierd Adrian Wojtasiewicz

All Rights Reserved© 1991 Kluwer Academic PublishersNo part of the material protected by this copyright notice may be reproduced orutilized in any form or by any means, electronic or mechanical,including photocopying, recording or by any information storage andretrieval system, without written permission from the copyright owner .

Printed in the Netherlands

To the memory of Jerzy Sfupecki

CONTENTS

Introduction ix

1 . Main assumptions, objectives, conditioningsix

2 . Intuitive foundations xv

Chapter I

The Axiomatic Theory TLTk of Label TokensI

Sec .I .i . Primitive concepts 1

Sec .I .2 . Label tokens and their equiformity3

Sec .I .3 . Concatenation 5

Sec .I .4 . Vocabulary 10

Sec .I .5 . Word tokens 12

Sec .I .6 . n-componential words ; length of word17

Sec .I .7 . Generalized concatenation 23

Sec .I .8 . Methodological remarks 29

Chapter II

The Axiomatic System TSCL of Simple Categorial Languages . 32

Sec .II .1 . The principal objectives of construction the

theory of syntactically categorial languages . 32

Sec .II .2 . Connections between TLTk and TSCL37

Sec .II .3 . Categorial indices and their indication42

Sec .II .4 . Expressions 46

Sec . I I .5 . Well-formed expressions 51

Sec .II .6 . Syntactic categories 61

Sec .II .7 . The fundamental theorems of the theory of

syntactic categories 66

viii CONTENTS

Sec . II .6 . The algorithm of checking the syntactic

correctness of expressions78

Chapter III

The Theory TSCe-L of Categorial w-languages90

Sec . III .1 . Introductory remarks 90

Sec . III . 2 . The foundations of TSCe-L95

Sec . II1 .3 . Operator expressions 97

Sec . III .4 . Well-formed expressions 103

Sec . III .5 . Fundamental theorems 109

Sec . III .6 . The algorithm of checking the syntactic

correctness of expressions120

Chapter IV

Dual Theories 132

Sec .IV .1 . The double ontological character of linguistic

objects and the biaspectual approach to

language 132

Sec .IV .2 . The theory TL Tp of label types139

Sec . I V .3 . Interpretation in TLTp of Tarski's axioms of

metascience 148

Sec . I V .4 . The theory TETp of expression types151

Sec .IV .S . The dual theory DTSCL 164

Sec .IV .6 . The dual theory DTSCe-L 171

Final Remarks 181

Annex 186

Notes . . . 233

References 237

INTRODUCTION

1 . Main assumptions, objectives and conditionings

1.1. The present book is concerned with certain problems

in the logical philosophy of language . It is written in the

spirit of the Polish logical, philosophical, and semiotic

tradition, and shows two conceptions of the syntax of

categorial languages : the theory of simple languages, i.e .,

languages which do not include variables nor the operators that

bind them (for instance, large fragments of natural languages,

languages of well-known sentential calculi, the language of

Aristotle's traditional syllogistic, languages of equationally

definable algebras), and the theory of w-languages, i .e .,

languages which include operators and variables bound by the

latter <for instance, the language of the functional calculus

and languages of the theories based on that calculus) .

The considerations on languages to be found in this book are

very general and, e .g ., do not depend on the notation used in

the recording of expressions in the languages under

consideration.

The influence of Polish logic is reflected in the strictly

formal manner of presentation, and in particular in the use of

the axiomatic method . It is commonly known that those important

factors dominated the enquiries of eminent representatives of

the Warsaw school of logic, such as Stanislaw Lesniewski, Jan

ix

x INTRODUCTION

Lukasiewicz, Alfred Tarski, Jerzy Slupecki, and others .

Reference to Polish semiotics manifests itself in the

categorial approach to language, i.e., the approach which in

the syntactic description of language takes into account the

general principles of Leigniewski's theory of syntactic

categories [1929,1930] in the version improved by Kazimierz

A jdukiewicz [1935] .

It is worth noting that Le6'niewski's theory of syntactic

categories was constructed under the influence of the ideas of

pure grammar, advanced by Edmund Husserl [1900,1901], and also

under the influence of the concept of logical type, for the

language of protothetics and ontology [1929,1930] . That theory

formulates general principles of the classification of

expressions into syntactic categories, its main objective being

the formulation of criteria of the meaningfulness,that is

well-formedness of expressions . In that spirit it was developed

or used by Tarski [1933], A jdukiewicz [1935], Innocenty M .

Bochei'iski [1947], Tadeusz Kubii ski [1958], Witold

Marciszewski (1977, 1978a], and others. For similar purposes,

although independently of Les'°niewski, a theory of syntactic

categories was presented and developed for the needs of

so-called combinatory logics by Haskell B. Curry [1961,1963] .

The categorial approach to language, used in the present

book, refers to A jdukiewicz's conception [1935] and also to his

ideas to be found in [1960], and is based on the method of

indexation of expressions, that is the method of assigning them

categorial indices and the resulting defining of their

the

INTRODUCTION xi

syntactic categories . That method makes it possible to apply

the algorithm which allows one to examine the well-formedness,

i.e ., syntactic correctness, of expressions Cin A jdukiewicz's

terminology, their syntactic connectedness>. The categorial

approach makes it possible to treat language as a language

generated by the appropriate categorial grammar, whose idea

goes back to A jdukiewicz [19357 .

The unquestionable achievements of Yehoshua Bar-Hillel [1950,

1953,1964], who shaped the concept of categorial grammar and

popularized Ajdukiewicz's conceptions, and thus made a great

contribution to the development of the foundations of

categorial grammars, and the equally unquestionable attainments

in that respect of Joachim Lambek [1958,19617 and the

continuators of these two prominent researchers, bear a certain

relation to the approach of the foundations of categorial

languages suggested in this book, even though they do no play

any essential role .

The philosophical trend of the theoretical reflections to be

found in the present book manifests itself in the biaspectual

treatment of language : as language of expression-tokens and as

language of expression-types, the latter being interpreted in

an abstract way. This is accompanied by the claim that it is

the expression-tokens, accessible to sensory cognition, which

are the basis of considerations on language, while the abstract

expressions (expression-types), of which the former are

representations, are derivative and definable constructs . This

approach is linked to the nominalistic Cconcretistic)

xii INTRODUCTION

standpoint in the philosophy of language, represented in

particular by such Polish logicians and philosophers as

S.Ledniewski, T.KotarbirSski, and also J .Slupecki (during the

last years of his life) .

These aspects which determine the main assumptions made in

this book, namely the logical, the linguistic, and the

philosophical aspects, result in a two-level formalization of

both theories of categorial languages, or theories of

categorial grammars . Formalization at the first level pertains

to languages of expression-tokens and yields the theory TSCL of

simple syntactically categorial languages <Chap .II) and the

theory TSCw-L of the w-languages of such expressions <Chap .

III) . Formalization at the second level pertains to

expression-types and yields, respectively, the dual theories

DTSCL and DTSCca-L (Chap .IV) . Those theories are expansions of

the relevant theories formalized at the first level.

One of the basic objectives is to demonstrate the mutual

relationships between theories formalized at these two

different levels .

1.2. Comprehension of the formalisms which yield the said

theories requires only an elementary knowledge of mathematical

logic in the broad sense of the term, but it assumes a certain

skill in the consistent application of the formal logical

apparatus . This is why, in order to make the Reader better

acquainted with the subject matter of the book we shall

formulate below <in Sec.2) the intuitions pertaining to the

INTRODUCTION

philosophical and linguistic aspects of the framework used and

to the formalisms themselves . They may prove useful in

explaining the main ideas underlying the two suggested formal

theories of the syntax of language . The understanding of the

book should also be facilitated by numerous verbal comments,

referring to intuition, in most cases accompanying the symbolic

recordings of axioms, definitions, and theorems <the proofs of

the latter are, in the overwhelming majority, transferred to

the Annex to make the perception of the main text easier) .

1.3. The thus outlined two-level system of treating both

theories of language has, as it were, its precedent in the

theory of algorithms formulated by A .A.Markov [1954], which,

however, is not an axiomatic theory . An endeavour to axiomatize

it was undertaken by Jerzy Slupecki . It was on his initiative

and under his guidance, and with reference to Markov's ideas,

that the axiomatic theory of concrete and abstract words was

formulated Csee C3 .Bryll and S .Miklos 11977]) . It was followed -

for other purposes, connected with the theory of syntactic

categories - by the axiomatic theory TLTk of label tokens,

constructed by the present author Csee 11981a,1983a]). These

goals coincided in the 1970's with the revived interests in

categoriel grammars, which had their semantic and philosophical

grounds. The fact was stressed by Witold Marciszewski [1978b]

and found manifestations in publications of such authors as

P.Geach 11970], M.J .Cresswell [1972,1976], D.Lewis Ci9721,

R.Montague (1970,19731, and A.Nowaczyk 11978] . That trend in

xiv INTRODUCTION

research had been anticipated by R.Suszko 11958,19601 .

The theory TLTk has become the basis for considerations on

syntactically categorial languages . It also forms the core of

all the theories of language presented in this book and

mentioned previously . It is shown in Chap . I . It is also an

outline of the intuitions connected with its formalization that

opens the preliminary presentation of the contents of the book

(Subsection 2.2) .

1.4 . But first a brief explanation is needed of the

relation which the English-language version bears to the Polish

original. The essential content of the book has remained

unchanged . Only the Introduction, the References and the Final

Remarks (which now form a separate item that follows Chap .IV)

have been expanded. This has been done for three essential

reasons: to make the book accessible to a much broader category

of Readers, to bring it closer to the world literature of the

subject, and to outline new solutions thematically connected

with the original version of the book. This, it seems, places

the book in a better perspective, especially in the light of

the recent achievements in the field of categorial grammars,

whose turbulent development originated in the 1950's and,

revived in the 1970's, has continued up till now . It must be

added here that this development has been influenced by the

still vital research on formal grammars in the spirit of Noam

Chomsky, going on incessantly since the publication of his book

119571 that marked a breakthrough in linguistics .

INTRODUCTION xv

It would not be possible here to indicate, with a view to

referring the present book to historical and present studies in

the sphere of categorial grammars, the differences and the

coincidences between this book and the said studies . This is

why I take the liberty to refer the Readers above all to the

entry "Categorial grammar" in the Dictionary of Logic edited by

Marciszewski [1981], the anthology bearing the same title and

edited by Buszkowski, van Benthem, and Marciszewski [1988], and

the selected publications by the leading contemporary students

of categorial grammars, W.Buszkowski [1982a-c,1984,1985,1986]

and J .van Benthem [1984,1986], which appeared when the Polish

version was being prepared for the press or after its

publication .

The essential part of the original version was ready in 1981,

and the main ideas of the underlying theoretical conceptions

were presented at the Conference on the History of Logic <see

[1981b]>, held in Cracow and dedicated to the scientific

production of Kazimierz Ajdukiewicz . The Polish version

appeared in 1985 . Professor Jerzy Slupecki, who suggested to me

the problems analysed in it, had died before the publication of

the English-language version . Let me, -therefore, dedicate it to

his memory .

2. Intuitive foundations

2.1. The controversy over the problem of the double

ontological nature of linguistic objects is linked to the well

xvi INTRODUCTION

known controversy over universals . The words written in the

preceding sentence are concrete physical objects perceivable by

sight . The second and the twentieth word which occur in the

first sentence in this paragraph, like the word occurring

within the quotation mark in the term "controversy" may be

treated as concrete egviform. representations of that name. That

name is often treated as the name of a certain abstraction,a

word type, that is the class of all the words which are

equiform with the word in the quotation marks . The class of

such concrete words is then an abstract word, or a certain

ideal entity. The controversy over the ontological nature of

the objects designated by the name "controversy" arises when we

want to decide which of the kinds of words denoted by that name

has original existence, and which has derivative existence .

Without making any decision in that matter we shall point to

the fact that the demonstrated duality in the interpretation of

words or expressions of language, which has become common

especially owing to the important study of A .Tarski [1933],

takes place in logic and semiotics . Note also that following

the distinction between token and type made by C.S.Peirce

[1931-1935] (see also Y.Bar-Hillel [1953,1954]) inscriptions,

words, expressions understood as concrete objects are included

in the class of tokens, and when understood as classes of

equiform tokens - are included in the class of types CR.Carnap

[1942] used, respectively, the terms events and designs.

In accordance with the well known division made by Ch .Morris

[1938] logical semantic analysis can be carried out in three

INTRODUCTION xvii

aspects : the syntactic, the semantic, and the pragmatic . Such

an analysis is sometimes applied to concrete and sometimes to

abstract linguistic objects. For instance,in syntactic studies,

when formulating rules of constructing coherent expressions we

make use of expression types, but if we want to examine the

grammatical correctness, i.e., syntactic connectedness, of

expressions and use for that purpose, e.g ., A jdukiewicz's

algorithm [1935] we make use of tokens . At the pragmatic level

linguistic analysis pertaining to the functioning of language

is linked to the use of expressions in definite contexts and

situations (see J .Pelc [1967], [197113 . Reference to the

numerous functions of linguistic expressions (descriptivity,

emotiveness, suggestiveness, decisiveness>, substantiation of

statements, and in particular theorem proving requires the use

of expression tokens . On the contrary, the functioning in

pragmatics of the concept of communication is based on the

concept of expression type (potential inscription) . It is the

same when it comes to such basic semantic concepts as

designation, denotation, truth, and meaning .

The said ambiguity in the use of linguistic inscriptions in

the analysis of language has also its broader dimension. This

is so because in the literature of the subject it is not always

clear whether expression type is understood as a class of

equiform tokens, sometimes there is even no reference to the

concept of equiformity even though the context indicates that

expression types are meant . W.Marciszewski claims that the

concept of type has several different interpretations and need

Xviii INTRODUCTION

not be understood in an abstract way .

Note in this connection that when speaking about the dual

ontological nature of linguistic objects we shall refer to

tokens and types in accordance with the distinction made by

Peirce .

The dual ontological nature of linguistic inscriptions and

the possibility of using them in two ways in semiotic analysis

results in the necessity of the biaspectual description of

language in the theoretical, logical conception of language, as

a language of expression tokens and a language of expression

types. Yet in formalizing a given conception of language we

must choose either tokens or types as the point of departure .

In the two approaches to the theory of language suggested

here we refer to the nominalistic Cconcretistic) interpretation

and construct those theories first as the theories TSCL and

TSCarL of languages of expression tokens (the first level> ; in

doing so we start from token (label token> as a primitive,

undefined, concept and several other concepts connected with

the former. When constructing those theories as theories of

expression types (the second level) we obtain the dual theories

DTSCL and DTSCCrL, respectively, in which the fundamental

concepts include such abstract concepts as label type, word

type,

constructs .

2.2. The theory TLTk of label tokens (Chap .I) is the point

of departure for the formalization of both conceptions of the

expression type, which are defined and derived

INTRODUCTION xix

syntax of language . In that theory four concepts appear as

primitive, undefined, whose meaning is explained by the

appropriate axioms . They are : the set Lb of all label tokens,

the relation ::~: of equiformity, the vocabulary V, distinguish

from the set Lb, and the relation C of concatenation.

The set Lb is the universe of an arbitrary but fixed simple

language W or w-language w-.x', which has tokens as its

expressions. The elements of Lb, called label tokens, are

concretes and may (but need not), be inscriptions . Label tokens

are any objects perceivable by sight, extensive in time and

space, from which by egvi formi ty one can obtain elements of

expressions of graphic languages. Out of label tokens we choose

the simplest words of the language . or w-X, namely the words

which form the vocabulary V of .LP or G)-X, and from them in turn

we form compound words of that language, the expressions,

including the well-formed expressions of which consist X or

w-X_ Label tokens also include the categorial indices which

make it possible to determine the syntactic categories of

expressions, to demonstrate their syntactic roles, and to

examine grammatical coherence . The existence of label tokens

and their varietes listed above follows from the axioms adopted

in the theories TSCL and TSCw-L .

The relation of equiformity holds between pairs of label

tokens . It is interpreted very broadly by imparting it a

pragmatic sense: it depends on the objective which we have in

view in order to obtain definite results . The equiformity of

label tokens may, but need not, be determined by physical

xx INTRODUCTION

similarity . For instance, for a visual artist, a graphologist,

or a printer the concrete words : ontology, ontology, ONTOLOGY

Glower case, lower case italics, capitals) are diversiform,

whereas from the point of view of a person who is learning the

meanings of words they may be

hand, in categorial analysis

treated as equiform . On the other

intended to examine the syntactic

correctness of expressions, for •instance of the tautological

sentence

The creative man is creative

the adjectives which occur in it may be treated as diversiform

in view of their enviroment : the first modifies a noun and is a

name-forming functor, while the second is independent and,

being a name, belongs to another syntactic category and has a

different categorial index (see H.Hi:C 11960,1967,19681) . Should

we resort to a metaphor, we might say that it is like with one

and the same deciduous tree in summer which differs from itself

in winter when it has no leaves . Note also that a word which

belongs to the vocabulary of a natural language or even a

formalized language analysed at the token level is equiform

with at least one word used earlier in the process of

communication and has a categorial index equiform with the

categorial index of the former, and hence also the same

grammatical category.

It is further assumed that the relation :-- of equiformity is

reflexive, symmetric, and transitive, without considering the

problem whether that postulate could be replaced by a less

INTRODUCTION xxi

rigorous one . It is also postulated that a label token which is

equiform with a word from the vocabulary of a language .L° or w-Z

is also a word from the vocabulary of that language .

The vocabulary V is used to generate the set W of all the

words of 2 or c,-X . That language is determined by the set S of

all its meaningful, i.e ., well-formed expressions, which is a

subset of W. The set W is generated from the vocabulary V by

the relation C of concatenation, which holds between any two

label tokens and the label token which is their concatenation .

W is the least set, which includes V and every label token which

is the concatenation of simple word tokens, i.e ., the ordinary

concatenation of two simple words from the vocabulary V, or the

generalized concatenation of more than two simple words from V.

At the token level the relation of concatenation need be a

function because two given label tokens may yield many equiform

concatenations. This is so because intuitively concatenation

may be understood as the relation of writing, on the right side

<e.g., in European ethnic languages) or on the left side

in Semitic languages), and at the same level, of a label token

equiform with the first label token in a given pair, of a label

token equif orm with the second label token in that pair. But it

may also be a relation of non-linear juxtaposition of two label

tokens which thus form a single label token. This may take

place, e .g ., in hieroglyphic script and in many exact sciences .

Upon the relation of concatenation we impose axiomatically

certain intuitive conditions related to A .Tarski's axioms of

metascience [1933] <see also e .g. H.Hi± [1957]), but in doing

xxii INTRODUCTION

so we have to bear in mind the fact that in our case that

relation is defined for tokens, and not for types . The axioms

of the theory TLTk guarantee, of course, that the words from

the vocabulary V of 2 or w-.L°, unlike the compound words from

the set W'.V of that language, are not concatenations of any two

label tokens .

The generalized concatenation of label tokens is an ordinary

concatenation or the concatenation of an ordinary concatenation

and a given label token. A compound word of .L° or c,- .e is thus

always a generalized concatenation of simple word tokens,

because it can be obtained by a single or multiple application

of ordinary concatenation to simple words from V . A compound

word is thus formed of a finite number of vocabulary words . The

number of such words which are parts of a given words is

defined as its length.

2.3. The second stage in the formalization of the theories

TSCL and TSCw-L of categorial languages ', and correspondingly

w-.i°, consists in the construction of the theory TETk of

expression tokens . Its presentation is to be found in Chap .II

where the theory TSCL is shown. Now the theory TETk is obtained

from TLTk by the introduction of two new essential concepts,

that of expression token and that of categorial index. Here are

the principal intuitions connected with the formalization of

that theory .

Categorial indices are used to carry out linguistic

categorial analysis in accordance with the ideas advanced by

INTRODUCTION xxiii

A jdukiewicz [1935,1960]. Such an analysis pertains

to any compound expression tokens that form the subset Ec of

the set W of word tokens . In accordance with the principle

originating from G.Frege [18791 every expression from the set

Eq- has the functor-argument structure - one can distinguish in

it its constituent called the main functor and its constituents

called the arguments of that functor .

The proper categorial analysis of a compound expression

token <the examination of its syntactic correctness) is

preceded by its functorial analysis : after having distinguished

in a given compound expression the main functor and its

arguments we continue the process of distinguishing the main

functor and its arguments with reference to all constituents of

a given expression which are compound expressions . By way of

example, the functorial analysis of the sentence "John ardently

loves Helen" is illustrated by two diagrams of a tree . In the

first, the bold lines indicate the main functors; in the second

the method of parsing used by Ajdukiewicz is taken into

consideration (see Fig.1 and Fig.1a below) .

exclusively

xxiv INTRODUCTION

John ardently e loves Helen John ardently loves Helen

/\John ardently loves Helen ardently loves John Helen

a .~ d+es +. . i y loves ardently laves

Fi g .1 Fig.1a

The categorial indices (types) by means of which we examine

the syntactical correctness of compound expression tokens are

label tokens of Lb . But they are not in the set W of the words

of Z or w-X. They are words from metalanguages of those

languages and, to use Buszkowski's terminology, are used for

the typisation, or determination of the syntactic categories,

of expressions in those languages and for the examination of

their grammatical correctness. For the formal syntactic

description of the categorial language .R ., or w-.L°, we introduce

a new primitive concept, namely that of auxiliary vocabulary I o

that is the set of all basic indices, which includes at least

one technical symbol, e.g ., slant /, colon :, comma „ bracket

( or ), etc . It is, of course, postulated, that the sets 1 0 and

V have no common elements. From the vocabulary 1 0 we generate,

by the relation C of concatenation, the set I o f all ca t egor i a l

INTRODUCTION xxv

indices, as the set W is generated from V . The set I thus

contains all basic indices from I and all indices of functors.0

The latter are generalized concatenations of basic indices. The

principles of the construction of the indices of functors of a

given categorial language .I° or w-.e are fixed by special rules .

For instance, of the basic indices s and n, used, respectively,

for the typization of sentences and names in such a language,

and the auxiliary symbol /, on the assumption that the

concatenation of label tokens provides for right-side linear

juxtaposition, we can form indices of functors of a

quasi-fractional form, e.g ., s/nn and s/nn//s/nn, which are,

respectively the index of the sentence-forming functor of two

name arguments, and the functor-forming functor, which has the

above functor as its only argument and forms such a functor .

The typization of definite words of X or w-Z is obtained by

the relation . of indication of indices of words, in other

words, the relation of typization . That relation is a new

primitive concept in the theory TETk. It is postulated that it

is a function which assigns to definite word tokens one index

token. It is also axiomatically assumed that words which are

equiform with a word that has an index have their indices

equiform with that of the said word .

From the set of those words which have indices, that is from

the domain D(e )o of the relation . , we distinguish the set E ofall expression tokens of . or w-X, and hence both the set E ofsall simple expressions of such a language, which are words of V

and have an index, and the set Ec (obtained from Es ) of all

xxvi INTRODUCTION

compound expression tokens of that language . The principles of

the formation, out of words that have indices, of compound

expressions of such a language are fixed by its syntactic

rules. In theoretical considerations those rules are replaced

by a single one, namely the one-to-one func tion p o f the

formation of compound expression tokens of .L° or w- .I° . The set Ec

is defined as the counter-domain D1Cp) of that relation .

The function p is the third and the last primitive concept

of the theory TETk . We associate with it the following

intuitions . It allows us to assign to any finite configuration

Cpo

, pi

pn), where n>i, of words of . or w-X that have

categorial indices, such a generalized concatenation of them

which in the language of TETk is recorded as

C6) p<po , P1 , . . . , Pn >

and termed a compound expression token of .L° or w-.L° consisting

of the words po,p1, . . .,pn . This is so because (6) is treated as

a substitute of any compound expression of such a language

which is constructed of the main functor po and its successive

arguments P1 , . . .,pn . In doing so we do not consider the formal

structure of such an expression and disregard the symbolism in

which it is recorded and how it is formed as the appropriate

generalized concatenation by rules valid in .L' or w-X. The

inscription <6) thus may denote various expressions of d° or

w-.k', built according to various rules, on the condition that in

each such expression the number of words is the same and that

among those words we can distinguish the main functor and its

INTRODUCTION xxvii

arguments. Moreover, such expressions need not be of one and

the same syntactic category; they may be sentences, names, and

functors. Further, (6) may also replace different but

synonymous expressions recorded in different notations (see

Sec .IL4) .

The theory TETk is enriched by new concepts, specific of

TSCL and TSCu -L, the most important of which is that of a

meaningful expression, i.e., welt-formed expression token . That

concept is defined differently for X and for w-Z .

2.4 . The theory TSCL describes any simple categorial

language .L° of expression tokens . Its welt-formed expressions

are elements of the set S, which in that theory is defined as

containing all simple expressions from Es and all those

compound expression tokens of .LP, which have the property that

both they and each constituent of theirs which is a compound

expression of that language satisfy the rule m which in a free

formulation states :

The categoriat index of the main functor of a compound

expression is obtained as a generalized concatenation from

the index of the expression, i .e., the index of the expression

which that functor forms together with its arguments, and

the indices of all successive arguments of that functor .

This definition of well-formed expression makes it possible

to describe an algorithm of examining the syntactic correctness

of linguistic expressions which comes close to A jdukiewicz's

algorithm [19351 . This is so because categorial analysis

consists in finding whether the rule m holds for every

constituent (which is an expression of . ') of a given expression

of .L° that consists of functors and arguments . If this is so,

then a given compound expression of X is in S and hence - is a

well-formed expression . By referring to the above example of an

expression whose functorial analysis was illustrated by Fig .i

and Fig.ia, we can easily check its well-formedness by

assigning categorial indices to its constituents. For that

purpose we can use the tree of categorial indices of that

expression shown in Fig .2 and Fig.2a below, where s and n, and

the quasi-fractional inscriptions are interpreted as explained

previously .

s/nn//s/nn s/nn

Fig.2

INTRODUCTION

s/nn//s/nn

Fig.2a

s%nn

It can also easily be seen that the set S of all well-formed

expressions of a categorial language .L° can be generated in this

way by the system

INTRODUCTION

GX . < Es , I o , e, m >,

which may be treated as a reconstruction of classical grammar,

whose idea goes back to A jdukiewicz (1935]. Such a grammar is

rigid (see W.Buszkowski (1986]), which is to say that a given

expression of .L° has only one index (type), and hence only one

syntactic category, assigned to it . It will be assumed about a

syntactically described categorial language X that it

unambiguous ly typizable (see W.Buszkowski (1986]) . Such an

assumption is quite natural because finitely typizable

languages, which we usually consider to be natural languages,

can always be disambiguated by the "separation" of words or

expressions to which a finite number of indices <types> is

assigned. That can be achieved if we provide them, for

instance, with the appropriate numerical indicators, letters,

or diacritical signs . Moreover, as has been said earlier, we

can very broadly interpret the concept of equiformity, which

substantiates the assumption made above .

The categorial nature of the language . is also connected

with the logical partition CtCS) of the set S of all

well-formed expressions of .Lp into non-empty and pairwise

disjoint syntactic categories . Syntactic categories are

determined by categorial indices and are sets of expressions

which have equiform categorial indices . In the family CtCS) of

all syntactic categories we can distinguish basic categories

and functoral categories . The former include all basic

expressions, i.e ., expressions with basic indices from Zo,

while the latter include func tors, i.e ., expressions with

XXX INTRODUCTION

functoral indices from the set I'I . The set of all basic0

expressions of .L° and the set of all its functors have no common

elements and the sum of those two sets cover all well-formed

expressions of d°.

The concept of syntactic category bears a certain relation

to the relations of repiaceabiiity of expressions, which is a

defined concept in TSCL . In the traditional interpretations a

syntactic category is the set of all expressions mutually

replaceable in arbitrary sentential contexts, or, more

generally, in all well-formed contexts . The traditional

definitions of that concept (see S .Led'niewski 11929], A .Tarski

11933] , K.Ajdukiewicz 11935] , I .M.Bocheiiski 11949]) imply

problems because they do not preclude a vicious circle, which

is due to the fact that the concepts of sentence or meaningful

(well-formed> expression and that of replaceability of

expressions function in them as undefined . Moreover, as was

stated by R.Carnap [1937] the principle of replaceability of

expressions in arbitrary well-formed contexts, assumed in the

traditional definitions of syntactic categories, may be

questioned (cf. also H.Hiz [1961]) . The correctness of such

definitions can be refuted by examples . In academic handbooks

of logic, e.g ., the definition of names guarantees to proper

names and personal pronouns belonging to the category of names,

whereas it can easily be seen that well-formed expressions,

respectively, a nominal and a sentential one (see J .Lambek

11958] >,

Big John, Big John is here,

RMODUCTION xxxi

on the replacement in them of the proper name by the personal

pronoun "he" yield meaningless <not well-formed) expressions

Big he, Big he is here .

On the other hand, for instance, both expressions "2=2" and

"2+2" are well-formed, but the latter is obtained from the

former by the replacement in it of the sentence-forming functor

_" by a functor belonging to another syntactic category,

namely the name-forming functor "+" .

Even though we deviate here from the traditional definitions

of the concept under consideration it remains in an important

connection with replaceabiixty . That connection is described in

Sec .IL7 by the fundamental theorems o f the theory o f syntactic

categories, at first intuitively discussed in Sec .ILi, where

the objectives of the construction of the theory of categorial

languages are outlined .

2.5. The theory TSCa-L describes an arbitrary but fixed

categorial w-language t,--w of expression tokens . Its well-formed

expressions belong to the set S, defined in a slightly

different way than in the theory TSCL, because its expressions

may include variables and operators that bind them. The

definition of S should imply all that as has been shown in

2.4, makes it possible to carry out the categorial analysis of

the expressions of (AY-.LP that could be correct from the point of

view of the theory of syntactic category . Such a definition

should accordingly enable us, in particular, to formulate an

algorithm of the examination of the syntactic correctness of

xxxii INTRODUCTION

expressions containing operators and variables bound by them .

The endeavours to formulate such an algorithm, undertaken

already by A jdukiewicz, as far as I know deviate more or less

from the general principles of the theory of syntactic

categories. An algorithm which refers to those principles is

described in Sec .III .6 .

In order to give a proper definition of a well-formed

expression in w-.e we add to the theory TETk new concepts, which

do not occur in TSCL. They are those of the set 0 of all

operators, the set Vr of all variables, and the relation Cfv)

of being a free variable in a given expression token. The first

two of them are ptimitive concepts in TSCa-L. The intuitive

sense which we impart to operators (elements of the set 0) and

to variables (elements of the set Vr) does not deviate from the

interpretation of these concepts in the exact sciences. To

simplify the matters we assume that one operator binds only one

variable while its scope is limited to only one expression

token . Operators and variables are treated as simple

expressions in w-X. Operators as such are elements of the

vocabulary of w-.L° and cannot be concatenations of any label

tokens, and moreover they can be assigned only one categorial

index each. Thus, for instance, we treat as an operator the

existential quantifier V or 3, and not the graphic symbol X or

3x, and we can assign it one categorial index (the same applies

to any inscription equiform with it) . This seems to be in

contradiction with the so-called typical ambiguity of operators

<see, e.g., R.Suszko 11958], 11964]), with their being treated

INTRODUCTION xxxiii

as categorial open functors <see D.R.Vanderveken [1976]), and

as context-dependent functors (see, e.g .,

[1989]) . For such an interpretation, graphically similar

quantifiers occurring, for instance, in the following two

arithmetical expressions and an expression of the sentential

calculus with quantifiers :

(i) X x < Y, X y x < Y, PCp -, -, P)

have different syntactic categories and hence also different

categorial indices . The first two because in Marciszewski's

[1989] interpretation the propositional functions x < y and

n x 5 y have different syntactic categories, and the thirdybecause, unlike the first two, it binds the variable

W.Marciszewski

p which

has the category of a sentence, and not of a name. This problem

can be eliminated as previously <see 2.2), either by treating

those quantifiers as diversiform because of their environment,

or by providing them with some diacritical signs which would

make it possible to treat them as diversiform .

Problems with assigning to the operators that bind variables

their syntactic categories have been known since the times of

Ledniewski [1929], Tarski [1933], and A jdukiewicz [1935] . In

the interpretation suggested here they will be treated as

functors of a special kind, which according to the context have

a definite categorial index each . That binding role of theirs

manifests itself in definite expressions of co-.LP called operator

expressions ; they are ordinary compound expression tokens con-

sisting of three constituents each: an operator, called the

xxxiv INTRODUCTION

main operator of that expression, a variable called indexical

variable, and an expression token called the scope of that

operator. Thus the main operator is the main functor of such an

expression, while the indexical variable and the scope of the

operator are its successive arguments .

The operator in an operator expression may be empty-binding

if in its scope it has no free variable or if it does not bind

any free variable in its scope (which is to say that no free

variable in its scope is equiform with its indexical variable .

The concept of free variable in a given expression is, of

course, defined by means of the relation Cfv>, namely as a

variable which bears the relation C fv) to that expression, if

it is a constituent of that expression and does not occur at

the same time in the scope of any operator in which it would be

equiform with its indexical variable .

The definition of the set S of all well-formed expression

tokens of w-X takes into consideration the degree of the

complexity of the expressions which are its elements . The

simplest well-formed expressions in that set are all its simple

expressions (including operators and variables> . The simplest

compound well-formed expression in that set is either i 0 a non-

-operator expression which is a compound expression token of

w- .LP consisting of simple expressions and such that its main

functor is not an operator and also such that it satisfies the

rule m, or 2' an operator expression such that the scope of its

main operator is the simplest non-operator expression

containing the variable which that operator binds, and such

INTRODUCTION

that this expression also satisfies the rule m (the index of

the main operator of that expression is a concatenation of the

index of that expression and the indices of the iridexical

variable of the main operator and that of its scope> .

More complex welt-formed non-operator expressions differ

from the simpler well-formed non-operator expressions only by

being constructed of at least one compound (operator or non-

-operator) well-formed expression, and more complex well-formed

operator expressions differ from simpler ones by the fact that

their scope is not a simplest well-formed compound expression.

Note that the set S thus includes all simple expressions of

w-X, all its non-operator expressions, and all its operator

expressions . Note also that every compound well-formed

expression of w-X and every constituent of such an expression

satisfy the rule m, which makes it possible to generate the set

S of w-X by the categorial grammar

Gw_X _ ( ES , I o , e, m >,

and also to describe the algorithm of the examination of the

syntactic connectedness of expression that resembles

A jdukiewicz's algorithm for expressions without operators that

bind variables .

When applying that algorithm we start from the parsing of a

given compound expression token of w-.' by carrying out the

functorial analysis of that expression as described under 2.3 .

If functorial analysis brings us to simple expressions (which

is to say, if the ends of the tree are simple expressions) of

xxxv

xxxvi INTRODUCTION

w-.L', then we find whether there is among them an empty-binding

operator . If it is not so, then we apply categorial analysis in

the proper sense of the term by assigning to every expression

obtained by parsing its categorial index and check whether

every compound expression thus obtained satisfies the rule m.

If this is so, then the given compound expression of w-l° is

well-formed. A precise description of that algorithm and

examples of its application are to be found in Sec .IIL6 .

Note also that if we apply the same designation with the

index s of sentences and propositional functions and if we

assign the same index n to names and name functions <and use

the same principles of forming index functors as in 2.3 and

2.4>, then the quantifiers which occur in the first two

expressions in Ci> have the index s/ns, whereas the quantifier

which occurs in the third expression in <i> has the index s/ss .

This is not at variance with the fact that the syntactic

category with the index s/ns covers such cases in indirect

speech as "thinks that" and "knows that", while the category

s/ss covers truth functors of two arguments each . This is so

because we do not adopt here the traditional definition of

syntactic category. By the way, in order to distinguish the

syntactic category closed sentences from that of

propositional functions in which one, two, etc., free variable

occur, we can accordingly use the typization: s,s,,s 2Petc . In

such a case the said existential quantifiers would have the

indices s/nsi, s/nsi and s/sisi , respectively, and the

universal quantifier in the second expression in Ci> would have

INTRODUCTION xxxvii

the index s/ns (see W.Marciszewski 1198913 . The above outlined2

algorithm obviously does work in such a case .

It must be emphasized at this point that all the properties

which make it possible to describe, in terms of TSCL, language

LP as a categorial language are applicable to language 63,-X as

well . In the theory TSCw-L all the analogues of the theorems

for well-formed expressions of £ hold, too .

2.6 . The formalization of the dual theories DTSCL and

DTSCw-L takes place at the second level, at which we are

interested in types and relations among them. Those

formalization pertain, respectively, to an arbitrary but fixed

simple language Y of expression types and an arbitrary but

fixed (L-language w-Y of such expressions .

All the linguistic elements of expression tokens of .L° or

w-.e, e.g ., label token, word token, expression token, well-

-formed expression token, variable token, operator token, and

object token which make it possible to carry out the categorial

analysis of expression tokens, and hence categorial index

tokens, have analogues in induced concepts which are elements

of expression types in .Lp or w-.L, namely label type, word type,

expression type, etc . and, for categorial analysis, index

types. All of them are defined as classes of abstraction of the

corresponding equiform tokens .

Likewise, all the relations holding among tokens have their

analogues in induced relations holding among types . The latter

are defined so that they hold among types if and only if

Xxxviii INTRODUCTION

relations with the same intuitive meaning hold among their

representations at the token level Crepresentatives of classes

of abstraction) .

All the properties of concepts grasped by the axioms,

definitions, and theorems of TSCL <or TSCw-L> which describe

categorially the syntax of .' <or w-.L°> of expression tokens pass

upon the induced concepts . This makes it possible to describe

categorially the syntax of a Cor w-Y> of expression types . We

find the full analogy between the syntactic concepts of the

languages in question analysed at two different levels . Since

in the syntactic description of language the assumption of the

existence of object tokens as the fundamental linguistic

elements served as the point of departure, we can conclude from

the above that a theoretical categorial syntactic description

of language does not require the preliminary assumption that

language is a language of abstract expressions, that is a

language of certain ideal entities . Hence in syntactic analyses

of categorial language we may abstain from making assumptions

about the existence of ideal linguistic objects .

This conclusion can be substantiated more strongly in the

light of recent studies carried out by the present author <see

Final Remarks> .

Acknowledgements

Professor Grzegorz Bryll was the first reader of the

typescript of the present book in its original version and

INTRODUCTION xxxix

checked the formal and factual correctness of my work . I owe

him my cordial thanks for that .

My thanks are also due to Professor Jan Srzednicki for his

initiative to have the book published in English . I am

particularly grateful to Professor Stanislaw J . Surma for his

efforts accompanying the publication of the, English version . I

also thank all those nameless friends who have contributed to

its appearance .